USB-C and BadUSB attacks: What you need to know

Apple announced the new MacBook with USB-C connector last Monday and already headlines are appearing linking it to known security issues, like BadUSB.
BadUSB is an attack that uses the way computers interface with the USB standard to try and load malware onto the machine. It's a longstanding issue with the universal serial bus in general, and nothing specific to Apple or the MacBook's implementation of USB-C. Throwing Apple and a hot new product in the headline gets attention, though! So, what's going on?